Problem 43 - Calculus with Concepts in Calculus

Question

Let \(f(x)=1+\sqrt{x+1}, g(x)=1+\sqrt{2-x}\), and \(h(x)\) \(=f(x)+g(x)\). Plot the graphs of \(f, g\), and \(h\) on the same calculator screen with viewing windows \([-10,10]\). Explain why the graph of \(h\) is so short.

Step-by-Step Solution

Answer

The graph of \(h(x)\) is short because \(h(x)\) is only defined on \([-1, 2]\), unlike the larger viewing window \([-10, 10]\).

Step 1: Determine the Domain of f(x)

The function \(f(x) = 1 + \sqrt{x+1}\) is defined for values of \(x\) such that \(x+1 \geq 0\). This implies \(x \geq -1\). The domain of \(f(x)\) is \([-1, \infty)\).

Step 2: Determine the Domain of g(x)

The function \(g(x) = 1 + \sqrt{2-x}\) is defined for values of \(x\) such that \(2-x \geq 0\). This means \(x \leq 2\). The domain of \(g(x)\) is \(( -\infty, 2]\).

Step 3: Find the Domain of h(x)

The function \(h(x) = f(x) + g(x)\) is defined at values where both \(f(x)\) and \(g(x)\) are defined. Therefore, \(h(x)\) is defined on the intersection of their domains, which is \([-1, 2]\).

Step 4: Plot the Functions

Using a calculator, plot \(f(x) = 1 + \sqrt{x+1}\), \(g(x) = 1 + \sqrt{2-x}\), and \(h(x) = f(x) + g(x)\) on the same graph with the viewing window \([-10, 10]\).

Step 5: Analyze the Graph of h(x)

Check the graph of \(h(x)\) and note that it extends only from \(x = -1\) to \(x = 2\), which is much shorter than the full viewing window \([-10, 10]\). This is because \(h(x)\) is only defined on \([-1, 2]\), which is a smaller interval than the viewing window.

Key Concepts

Domain of a Function

In mathematics, the domain of a function is the set of all possible input values (usually represented by \(x\)) that allow the function to work without any issues. Each function has specific conditions for which it is defined. For example, for the function \(f(x) = 1 + \sqrt{x+1}\), it is essential that the expression under the square root, \(x+1\), is non-negative. This is because the square root of a negative number is not a real number. Thus, we solve the inequality \(x+1 \geq 0\) to find out that \(x \geq -1\). Hence, the domain of the function \(f(x)\) is \([-1, \infty)\).
Similarly, another function like \(g(x) = 1 + \sqrt{2-x}\) requires that \(2-x \geq 0\). Solving this inequality, we find that \(x \leq 2\), giving us the domain \(( -\infty, 2]\) for \(g(x)\). Understanding domains is crucial because they tell us the range of \(x\) values for which the function can produce real, sensible outputs.

Intersection of Domains

When we talk about the intersection of domains, we refer to the common set of \(x\) values that satisfy the domain conditions of two or more functions. For functions \(f(x)\) and \(g(x)\), it's the set of \(x\) values where both functions are defined. For the function \(h(x) = f(x) + g(x)\), its domain depends on both \(f(x)\) and \(g(x)\).

The domain of \(f(x)\) is \([-1, \infty)\), and the domain of \(g(x)\) is \((- \infty, 2]\). To find the domain of \(h(x)\), we must take the intersection of these two domains, which is where both conditions are satisfied. Graphically or mathematically, this intersection is found by taking the overlap, resulting in the domain \([-1, 2]\) for \(h(x)\). This is smaller than either of the individual domains, as it needs to adhere to the constraints from both functions concurrently.

Graphing Functions

Graphing functions offers a visual understanding of how a function behaves over its domain. When plotting multiple functions, like \(f(x)\), \(g(x)\), and \(h(x)\), within a set viewing window (such as \([-10, 10]\)), it's crucial to remember the domain restrictions of each function.

• \(f(x) = 1 + \sqrt{x+1}\) appears on the graph starting at \(x = -1\) and extending to the right.
• \(g(x) = 1 + \sqrt{2-x}\) shows up on the graph starting at \(x = 2\) and extending to the left.
• \(h(x) = f(x) + g(x)\) can only be graphed from \(x = -1\) to \(x = 2\), as constrained by the intersection of the domains of \(f(x)\) and \(g(x)\).

When graphed on the same axes, \(h(x)\)'s graph looks much shorter than the others because it is only defined and visually represented in the interval \([-1, 2]\). The viewing window, \([-10, 10]\), is much larger, so the graph of \(h(x)\) occupies a relatively small portion. This highlights how domain restrictions directly influence the appearance and extent of a function's graph.